Differential and geometric structure for the tangent bundle of a projective limit manifold

Differential and Geometric Structure for the Tangent Bundle of a Projective Limit Manifold.

GEORGE N. GALANIS(*)

ABSTRACT- The tangent bundle of a wide class of Fréchet manifolds is studied he- re. A vector bundle structure is obtained with structural group a topological subgroup of the general linear group of the fiber type. Moreover, basic geo- metric results, known form the classical case of finite dimensional manifolds, are recovered here: Connections can be defined and are characterized by a generalized type of Christoffel symbols while, at the same time, parallel di- splacements of curves are possible despite the problems concerning differen- tial equations in Fréchet spaces.

Introduction.

The study of infinite dimensional manifolds and bundles has been se- riously developed the last decades with a number of applications which surpass the borders of Differential Geometry. However, several que- stions remain open, especially for the case of infinite dimensional mani- folds whose model are not of Banach type.

To be more precise, two problems seem to be of fundamental impor- tance: The lack of a general solvability theory of differential equations in non-Banach topological vector spaces and the pathological structure of general linear groups in this framework. Both, have serious impacts even at the first steps of the study of the corresponding manifolds.

Concerning, for example, the tangent bundle TM of a smooth mani- fold M modeled on a topological vector space F of the aforementioned (*) Indirizzo dell’A.: Hellenic Naval Academy, Section of Mathematics, Hatzi- kyriakion, GR-185 39 Piraeus, Greece. E-mail: ggalanisHsnd.edu.gr

type, we cannot even be sure for the existence of a vector bundle structu- re on it. Indeed, the classical folklore cannot be patterned here since the general linear groupGL(F), that serves as the structural group in the fi- nite dimensional case, does not admit a reasonable Lie-group structure.

On the other hand, even if this obstacle is overcame in some way, the study of many basic geometric entities of M (e.g. connections, parallel trans- lations, holonomy groups, etc.) seems also to be under question due to pro- blems with the solvability of the involved differential equations.

Despite the abovementioned difficulties, the study of certain types of infinite dimensional manifolds modeled on non-Banach topological vec- tor spaces is in several cases inevitable because of the extended use of such type of manifolds in modern Differential Geometry and Mathema- tical Physics. This need led a number of authors (e.g. [8], [10], [11]) to study certain types of infinite dimensional manifolds using, in most of the cases, rather algebraic approaches.

In this paper we study a certain sub-category of infinite dimensional non-Banach manifolds: Those that are modeled on Fréchet (i.e. locally convex, metrizable, Hausdorff and complete) spaces. Taking advantage of the fact that any Fréchet space can be realized as a projective limit of an appropriate sequence of Banach spaces, we focus further our atten- tion on those Fréchet-modeled manifolds M that are, correspondingly, projective limits of Banach manifolds. In a previous work M.C. Abbati and A. Manià ([6]) worked on projective limits of manifolds by using a pure algebraic approach. No real differential structure is determined on them and several fundamental notions are treated only through the pro- perties of projective limits. So, the tangent bundle of such a manifold M4lim

J Mⁱwas determined to be the projective limit of its counterparts on the factors:TM»4lim

J TMⁱ without any other assumptions referring or providing any differential or vector bundle structure. As a result, the obtained space is only a topological one since even the local triviality of it cannot be ensured. Analogous difficulties one faces also in several other subjects (differentiability of mappings, additional structures) thus the corresponding geometric properties are set also under question.

Throughout our note, based on a new approach, we overcome the abo- vementioned problems proving not only that any projective limit mani- foldM4lim

J Mⁱcan be endowed with a differential structure in the clas- sical way but also that a vector bundle structure can be defined on its tangent space TM having a new structural group that replaces the pa-

thologicalGL(F). On the other hand, the geometric study of this bundle goes surprisingly far: Connections can be defined and are characterized by a generalized type of Christoffel symbols, parallel translation along curves of M is succeeded and a study of the corresponding holonomy groups can be attempted.

1. Preliminaries.

In this first Section we introduce the basic notions and all the preli- minary material needed for a complete and self-contained presentation of the paper.

As already discussed in the Introduction, we are going to approach and study infinite dimensional manifolds modeled on Fréchet spaces based on the algebraic, but also compatible in several cases with diffe- rential tools, notion of projective limits. Our motivation was, at least par- tly, the fact that every Fréchet spaceFcan be always realized as a pro- jective limit of a sequence of Banach spaces ]Eⁱ;r^ji(^i,jN: F`lim

J Eⁱ (see [9]). Using this interpretation we have already studied some funda- mental problems onF, e.g. solution of linear differential equations ([1]), Floquet-type theorems in corresponding manifolds ([2]), etc.

On the other hand, this approach gives us the opportunity to partially overcome one of the main drawbacks in the study of Fréchet spaces with several reflections in Differential Geometry. Namely, it is well known that the general linear group GL(F) of F, which keeps a fundamental role in a number of issues of Analysis and Differential Geometry, is very difficult to be handled within the framework of Fréchet spaces and ma- nifolds since it cannot be endowed not only with a smooth Lie group structure but even with a reasonable topological group structure. Here we are going to replace GL(F) with a topological and, in a generalized sense smooth Lie, group defined as follows:

H₀(F)»4

m

^»

n

(1.1)

This is a topological group, since it coincides with the projective limit of the Banach Lie groups

H₀ⁱ(F)»4

m

^»

n

via the identification

(fⁱ)_iN`

(

)

H₀(F) may also considered as a«generalized» Lie groupif we thought of it as embedded in the Fréchet space

H(F)»4

m

^»

n

Similarly, the natural continuous mapping e: H(F)OL^{(F) : (fi)}iNOlim

J fⁱ, (1.2)

can be also thought of as a generalized smooth mapping if we restrict it onto H₀(F).

2. A certain type of Fréchet manifolds and the corresponding tan- gent bundles.

Based on the thoughts presented in the previous Section, we study here a certain type of Fréchet manifolds: Those which can be obtained as projective limits of Banach corresponding manifolds.

DEFINITION 2.1. Let ]Mⁱ;f^ji(^i,jN be a projective system of smooth manifolds modeled on the Banach spaces ]Eⁱ(^iN respectively. We as- sume that:

(1) The models ]Eⁱ(ⁱN form a projective system with connecting morphisms ]r^ji:E^jKEⁱ; jFi( and limit the Fréchet space F4 4lim

J Eⁱ.

(2) For any element x4(xⁱ)_iNM4lim

J Mⁱ there exists a family of charts](Uⁱ,cⁱ)(iNof Mⁱ’s such that the limitslim

J Uⁱ, lim

J cⁱcan be defined and the sets lim

J Uⁱ, lim

J cⁱ(lim

J Uⁱ) are open in M, F respec- tively.

Then the limit M4lim

J Mⁱ is called a PLB-manifold.

Under the assumptions of the previous Definition, we easily check that a PLB-manifold M turns to be a Fréchet manifold modeled on F.

The corresponding local structure is fully determined by the charts (limJ Uⁱ, lim

J cⁱ). The differentiability of mappings involved can be either this of J. A. Leslie ([5]) or that of A.Kriegl-P.Michor ([7]).

Characteristic examples of such type of manifolds are the PLBL- groups presented in [3] as well as the space of infinite jets of a Banach modeled manifold.

Using now the notion of projective limits and a new technique, we are going to study the tangent bundle of a smooth Fréchet manifold of the above type. This study in the general case seems to encounter serious obstacles from the very beginning. Indeed, although the definition of a smooth manifold structure on the tangent bundle can be finally achieved, any try to endow it with a vector bundle structure following the classical procedure is doomed to failure due to the pathological structure of gene- ral linear groups of Fréchet spaces. However, focusing on a PLB-mani- fold M4lim

J Mⁱ we obtain the next, necessary for the sequel, result.

PROPOSITION 2.2. The tangent bundles ]TMⁱ(^iN form a projective system with limit set-theoretically isomorphic to TM: TMC Clim

J TMⁱ.

PROOF. For any indexes (i,j), with jFi, we define the mapp- ing:

q^ji:TM^jKTMⁱ: [a,x]^jO[f^jiia,f^ji(x) ]ⁱ,

where the brackets [Q,Q]^j, [Q,Q]ⁱ stands for the equivalence classes of TM^j, TMⁱ with respect to the classical equivalence relations

aA^xb ` a( 0 )4b( 0 )4x and a8( 0 )4b8( 0 ).

(2)

Here by a8 we denote the first derivative of a:

a8:RKTM^j:tO[da(t) ]( 1 ).

We may check that eachq^jiis well-defined since two equivalent curvesa, b on M^jwill give

(f^jiia)( 0 )4(f^jiib)( 0 ),

(f^jiia)8( 0 )4df^ji(a( 0 ) )(a8( 0 ) )4df^ji(b( 0 ) )(b8( 0 ) )4(f^jiib)8( 0 ), where by df^ji:TM^jKTMⁱ we denote the first differential of f^ji.

Moreover,q^ikiq^ji4q^jk(jFiFk) holds, as a consequence of the cor- responding relations for]W^ji(^i,jN. Therefore,]TMⁱ;q^ji(^i,jNis a pro- jective system and lim

J TMⁱ can be defined.

Let now fⁱ:MKMⁱ be the canonical projections of the limit M4

4lim

J Mⁱ. Then, we set

Qⁱ:TMKTMⁱ: [a,x]K[fⁱia,fⁱ(x) ]ⁱ (iN) (3.12)

and we readily verify that q^jiiQ^j4Qⁱ holds for any jFi. As a result Q4lim

J Qⁱ:TMKlim

J TMⁱ: [a,x]K( [fⁱia,fⁱ(x) ]ⁱ)_iN can be defined. This is a one to one mapping because Q( [a,x] )4 4Q( [b,x] ) gives

dfⁱ(a( 0 ) )(a8( 0 ) )4(fⁱ_ia)8( 0 )4(fⁱ_ib)8( 0 )4dfⁱ(b( 0 ) )(b8( 0 ) ), and, therefore, a8( 0 )4b8( 0 ) since a4lim

J(fⁱia) and b4 4lim

J(fⁱib).

Moreover, Q is surjective since if a4( [aⁱ,xⁱ]ⁱ)iN is an arbitrarily chosen element of lim

J TMⁱ we obtain:

[f^jiia^j,f^ji(x^j) ]ⁱ4[aⁱ,xⁱ]ⁱ, for jFi. (3)

As a result, x4(xⁱ)M4lim

J Mⁱ and if ](Uⁱ,cⁱ)(^iN is a system of charts onMthroughx, as in Definition 2.1 and ((pⁱ)²¹(Uⁱ), lim

J Tcⁱ) the corresponding charts of]TMⁱ(iN (pⁱ: TMⁱKMⁱdenoting the classical projections of TMⁱ ), we obtain:

( (cⁱif^jiia^j)( 0 ),Tcⁱ( (f^jiia^j)8( 0 ) ) )4( (cⁱiaⁱ)( 0 ),Tcⁱ( (aⁱ)8( 0 ) ) )¨ (r^ji( (c^jia^j)( 0 ) ),Tcⁱ(Tf^ji( (a^j)8( 0 ) ) ) )4( (cⁱiaⁱ)( 0 ),Tcⁱ( (aⁱ)8( 0 ) ) )¨

(r^ji( (c^jia^j)( 0 ) ),r^ji( (c^jia^j)8( 0 ) )4( (cⁱiaⁱ)( 0 ), (cⁱiaⁱ)8( 0 ) ).

As a result,u4( (cⁱ_iaⁱ)( 0 ) )_iN, v4( (cⁱ_iaⁱ)8( 0 ) )_iN are elements of F4lim

J Eⁱ. If we consider the curvehof Fwithh(t)4u1tQv, as well as the corresponding one a of M with respect to the chart (U4lim

J Uⁱ, c4lim

J cⁱ), we may check that

(fⁱia)( 0 )4fⁱ(x)4xⁱ4aⁱ( 0 ),

(fⁱia)8( 0 )4( (cⁱ)²¹irⁱih)8( 0 )4(Tcⁱ)²¹( (rⁱih)8( 0 ) )4 4(Tcⁱ)²¹(rⁱ(v) )4(Tcⁱ)²¹( (cⁱ_iaⁱ)8( 0 ) )4 4(aⁱ)8( 0 ),

for anyi,jwith jFi. Therefore, the curves fⁱia,aⁱ are equivalent on Mⁱ (iN) and Q( [a,x] )4( [aⁱ,xⁱ]ⁱ)_iN4a.

We have proved in this way that TM, lim

J TMⁱ are isomorphic sets with respect to the mapping Q. r

Based now on the previous result, we may proceed with the definition of a vector bundle structure on the tangent bundle TM of M.

THEOREM 2.3. TM admits a Fréchet type vector bundle structure over M with structural group H0(F).

PROOF Let ](U_n4lim

J U_nⁱ,c_n4lim

J c_nⁱ)(^nIbe an atlas of M consi- sting of charts that can be obtained as projective limits in the form explained in Definition 2.1. Then, the atlas](U_nⁱ,cin)(^nI of Mⁱ can be used to define a vector bundle structure of fiber typeEⁱonTMⁱover Mⁱ in the classical way with trivializations:

tin: (pⁱ)²¹(U_nⁱ)KU_nⁱ3Eⁱ: [a,x]ⁱO(x, (cⁱ_nia)8( 0 ) ); nI.

Taking into account that the families]q^ji(^i,jN,]W^ji(^i,jN,]r^ji(^i,jN are connecting morphisms of the projective systems TM4lim

J TMⁱ, M4lim

J Mⁱ, F4lim

J Eⁱ respectively, we check that the projections ]pⁱ(^iN satisfy

W^jiip^j4pⁱiq^ji (jFi) and the trivializations ]tni

(^iN (W^ji3r^ji)_itnj

4tni

iq^ji (jFi).

Therefore, p4lim

J pⁱ:TMKM exists and is a surjective mapping, tn4lim

J tin:p²¹(U_n)KUn3F (nI)

are smooth, as projective limits of smooth mappings (see also [3] for a de- tailed proof), and pr₁itn4p, if pr₁ denotes the projection to the first factor.

Moreover, the restrictions of tn to the fibers p²¹(x) are linear iso- morphisms since t_n,x»4pr₂it_nNp²¹(x)4lim

J(pr₂it_nⁱ N(pⁱ)²¹(x)).

Concerning, finally, the transition functions ]T_nm4tn,xi itm21,x

(^n,mI, we see that they can be considered as taking values in the group H0(F), since Tnm4eiTnm* , where ]Tnm* (^n,mI are the smooth mappings

T_nm* : U_nOU_mKH0(F) :xO(pr2itⁱ_nN^(pi)21(x)i(tⁱ_m)²¹N^(pi)21(x))_iN and e the natural inclusion e : H⁰(F)KL^{(F) : (fi)}iNOlim

J fⁱ.

As a result, TM admits, indeed, a vector bundle structure over M with fibers of type F and structural group H₀(F). r

3. Projective limits of connections.

Here we are going to study some basic geometric properties of the in- finite dimensional manifolds presented in Section 2. To be more precise, we will focus on the characterization of connections via Christoffel sym- bols as well as on the possibility of having parallel translation of curves with respect to a connection. Both issues are under question in the gene- ral case of Fréchet manifolds due to intrinsic difficulties of their models:

The lack of a general solvability theory for ordinary differential equa- tions dooms to failure any try to apply the classical theory in order to ob- tain parallel translations. On the other hand, the fact that the space of li- near mappings between Fréchet spaces does not remain in the same ca- tegory, sets serious obstacles in the relationship between connections and Christoffel symbols.

However, if we restrict our study to those Fréchet manifolds that can be obtained as projective limits of Banach manifolds, in the sense describ- ed in the previous Sections, we may recover a great number of funda- mental results of the theory of finite dimensional manifolds and connections.

To this end we consider M4lim

J Mⁱ a PLB-manifold and a corre- sponding atlas](U_n4lim

J U_nⁱ,cn4lim

J cin)(^nIas in Definition 2.1. We denote by](p²¹(U_n),Fn)(^nI the corresponding local structure of the tangent bundle TM with:

Fn:p²¹(U_n)Kcn(U_n)3F : [a,x]O(c_n(x), (c_nia)8( 0 ) ); nI, and by](p21TM(p²¹(Un) ), Cn)(^nIthe obtained atlas on the tangent bun- dleT(TM) ofTM. Clearly, all these charts can be realized as projective limits of their counterparts on the factors TMⁱ,T(TMⁱ): Fn4lim

J Fni, Cn4lim

J Cni. If now

Dⁱ:T(TMⁱ)KTMⁱ

is a connection on the manifoldMⁱ, in the sense of J. Vilms ([12]), for any iN, and we assume that they form a projective system:

TW^jiiD^j4DⁱiT(TW^ji), jFi, (3.3)

then the corresponding limit D4lim

J Dⁱ: T(TM)KTM, which can be naturally defined as a smooth mapping according to the results of the

previous Section, seems to be far from being called a connection of M due to the difficulties emerged in the study of the corresponding local forms:

v_n:c_n(U_n)3FKL^(F,F)

defined by relations D_n(y,u,v,w)4(y,w1vn(y,u)(v) ), where D_n stands for the restriction of Donto the local charts of T(TM) and TM:

Dn4FniDiC21n (nI). Indeed, since the space of linear continuous mappings of any Fréchet space does not remain into the same category of topological vector spaces, it is not possible to realize the forms]vn( as projective limits of their counterparts obtained by the connectionsDⁱ and, therefore, as smooth maps. However, making again use of the space H(F)»4 ](fⁱ)iN

»

i41 Q

L^{(Ei) : lim}

J fⁱ exists(, defined in Section 1, we may overcome these difficulties obtained the following main result.

THEOREM 3.1. The projective limit D4lim

J Dⁱ of a system of con- nections on the PLB-manifold M is a connection characterized by a ge- neralized type of Christoffel symbols with values into the Fréchet space H(F).

PROOF We observe first that relations (3.3) and the fact that the charts ](p²¹(U_n),Fn)(^nI, ](p_TM²¹(p²¹(U_n) ), Cn)(^nI have been cho- sen to be projective limits of corresponding charts onTMⁱ and T(TMⁱ) respectively, ensure that the mappings]D_n(^nI can be also realized as projective limits. Indeed, if ]D_nⁱ(^nI are the corresponding restrictions of the connectionsDⁱ on the charts ofMⁱ(iN) andr^ji:E^jKEⁱ (jFi) the connecting morphisms of the model F4lim

J Eⁱ, we check that:

(r^ji3r^ji)iD_n^j4(r^ji3r^ji)i(F_n^j iD^ji(C_n^j)²¹)4 4Fin

iTf^jiiD^ji(C_n^j)²¹4Fni

iDⁱiT(Tf^ji)i(C_n^j)²¹4 4Fin

iDⁱi(C_nⁱ)²¹i(r^ji3r^ji3r^ji3r^ji)4D_nⁱi(r^ji3r^ji3r^ji3r^ji).

Similarly, we prove that (rⁱ3rⁱ)iD_n4D_nⁱi(rⁱ3rⁱ3rⁱ3rⁱ), for any iN, where rⁱ:FKEⁱ are the canonical projections of F. As a re- sult, Dn4lim

J Dni for each nI.

Concerning now the local forms v_n: c_n(U_n)3FKL^{(F,F) of D as} well as the corresponding onesvni:cni(U_nⁱ)3EⁱKL^{(Ei,Ei) ofDi’s (i} N), we cannot demand of them to be related by a similar manner, since

the projective limit of the spaces ]L^(Ei,Ei)₍iN can not be defined.

However, we may check that for anyy4(yⁱ)cn(U_n) andu4(uⁱ)F, the family (vni(xⁱ,yⁱ) )iN belongs to the Fréchet space H(F). Indeed, relations (r^ji3r^ji)iD_n^j4D_nⁱi(r^ji3r^ji3r^ji3r^ji), jFi, obtained abo- ve, give rise to

(r^ji3r^ji)(D_n^j(y^j,u^j,v^j,w^j) )4D_nⁱ(r^ji(y^j),r^ji(u^j),r^ji(v^j),r^ji(w^j) ) ` (r<sup>ji</sup>3r<sup>ji</sup>)(y<sup>j</sup>,w<sup>j</sup>1vnj(y<sup>j</sup>,u<sup>j</sup>)(v<sup>j</sup>) )4(y<sup>i</sup>,w<sup>i</sup>1vin(y<sup>i</sup>,u<sup>i</sup>)(v<sup>i</sup>) ) `

r^jiivnj(y^j,u^j)4vin(yⁱ,uⁱ)_ir^ji. As a result, we can define the mapping

v* :n cn(U_n)3FKH(F) : ((yⁱ), (uⁱ) )O(vⁱ_n(yⁱ,uⁱ) )_iN

which is smooth, as the projective limit of the smooth (Banach) local forms

vA_nⁱ:cⁱ_n(U_n)3EⁱKHⁱ(F) : (y,u)O(v_n^k(r^ik(y),r^ik(u) ) )_{k41 , 2 ,}R,i. These generalized local forms ofDare connected to the classical ones via the relation

vn4eivn* ,

where e stands for the natural inclusion e :H(F)OL^{(F) :} (fⁱ)_iNOlim

J fⁱ, defined in Section 1. This exactly is the fact that allows us to conclude that the local forms]vn(^nI are smooth and that Dis a connection of M under J.Vilms’ point of view ([12]).

Concerning finally the Christoffel symbols ]Gn:cn(U_n)K KL^(F,L^{(F,F) )}₍nI that characterize the above connection we check that

Gn(y)[u]4vn(y,u)4(eiv* )(yn ,u)4lim

J(vⁱ_n(yⁱ,uⁱ) )4 4lim

J(G_nⁱ(yⁱ)[uⁱ] ), for anyy4(yⁱ)cn(U_n) andu4(uⁱ)F. Therefore, these Christoffel symbols may be considered as taking values into the Fréchet spaceH(F) with respect to the inclusion e. r

REMARK 3.2. It is worth noticing here that in the case where the initial connections]Dⁱ( are linear the same holds for the correspond-

ing limit connection D4lim

J Dⁱ. Indeed, the linearity of Dⁱ’s is equiva- lent to the linearity of the second variable of the local forms ]vin: cin(U_nⁱ)3EⁱKL^(Ei,Ei)₍ which, in turns, gives rise to the corre- sponding property for the generalized local forms ]vA_nⁱ:cin(U_n)3EⁱK KHⁱ(F)(. Taking into account that the notion of projective limits is com- patible with linear structures, we conclude that the same holds for ]v* :_n c_n(U_n)3FKH(F)( and, thus, connection D is also linear.

Taking now one step further, we are going to study the possibility of having parallel translations of curves of the PLB manifold M4lim

J Mⁱ with respect to a linear connectionD4lim

J Dⁱ of the above type. In the general case of arbitrarily chosen Fréchet manifolds and connections, this basic geometric property fails to be recovered. Indeed, such a proce- dure demands the existence of horizontal lifts with respect to the chosen connection, thus the solution of linear differential equations on the fiber type. However, this is not possible on Fréchet spaces where even trivial linear differential equations may not be solved or may have more than one solution through the same initial condition (see for more details and corresponding counterexamples [4]).

By restricting our attention to the case of Fréchet manifolds and con- nections obtained as projective limits, we may overcome the previous diffi- culties avoiding to interfere at all with differential equations. Namely:

PROPOSITION 3.3. Let M4lim

J Mⁱbe a PLB manifold, as defined in Section2,and D a linear connection on M that can be realized as a pro- jective limit of connections: D4lim

J Dⁱ. Then, for any smooth curve a : [ 0 , 1 ]KM and for any element uTM, there exists a unique hori- zontal lift ja,u of a on TM satisfying the initial condition ja,u(0)4u.

PROOF. We observe first thatacan be thought of as the projective li- mit of the curves aⁱ»4fⁱia : [ 0 , 1 ]KMⁱ (iN), where fⁱ: MKMⁱ are the canonical projections of the limitM4lim

J Mⁱ. On the other hand, the elementu of TM4lim

J TMⁱ will have the form:u4(uⁱ), with uⁱ MⁱandTf^ji(u^j)4uⁱ,jFi. Taking into account that each manifoldMⁱis of Banach type, where the aforementioned problems concerning the sol- vability of linear differential equations are not longer present, we obtain, using the classical folklore, a horizontal liftjaiⁱ,uⁱ: [ 0 , 1 ]KTMⁱ through

( 0 ,uⁱ), for any iN. Moreover, and for any indexes jFi, we check that:

pⁱi(Tf^jiijaj^j,u^j)4f^jiip^jijaj^j,u^j

4f^jiia^j4aⁱ, DⁱiT(Tf^jiij_a^jj_,u^j)i¯4Tf^jiiD^jiTj_a^jj_,u^ji¯4Tf^jii040 , where¯stands for the basic vector field ofR. As a result, the vector field Tf^jiijaj^j,u^jof Mⁱ is also a horizontal lift of the smooth curve aⁱ satis- fying the initial condition (Tf^ji_ij_aj^j,u^j)( 0 )4Tf^ji(u^j)4uⁱ. Thus, having in mind the uniqueness of horizontal lifts on Banach manifolds, we con- clude that

Tf^jiijaj^j,u^j

4jiaⁱ,uⁱ, jFi. As a result,ja,u»4lim

J jiaⁱ,uⁱ: [ 0 , 1 ]KTM can be defined. This the de- sired horizontal lift of the smooth curve a on TM since

pij_a,u4lim

J(pⁱijⁱ_aⁱ_,uⁱ)4lim

J aⁱ4a, DiTja,ui¯4lim

J(DⁱiTj_aiⁱ,uⁱ

i¯)4lim

J 040 , ja,u( 0 )4(jⁱ_aⁱ_,uⁱ( 0 ) )4(uⁱ)4u.

The uniqueness of ja,u can be proved following analogous thoughts and observing that ifj* is a second horizontal lift ofa onTMsatisfying the same initial condition ( 0 , (uⁱ) ), then the factorsTfⁱija,u,Tfⁱij* would coincide as horizontal lifts of fⁱia on TMⁱ through the same ini- tial condition ( 0 ,uⁱ). r

A direct consequence of the previous Proposition is the following main result.

THEOREM 3.4. If M4lim

J Mⁱ is a PLB manifold and D4lim

J Dⁱ a linear connection on it, then, for any smooth curvea : [ 0 , 1 ]KM a pa- rallel translation

ta: p²¹(a( 0 ) )Kp²¹(a( 1 ) ) :uOja,u( 1 ) can be defined. r

REMARK 3.5. It is also remarkable here that the corresponding ho- lonomy group Fx4 ]ta, aC^Q( [ 0 , 1 ],M) and a( 0 )4a( 1 )4x( is

now a subgroup of the topological group H₀(F)and not of the(patholo- gical in our framework) general linear group GL(F)as in the classical case. Indeed, as we readily verify using the thoughts presented in the proof of Proposition3.3,t_a4lim

J t_aⁱⁱholds for any smooth curvea ofM if by taiⁱ we denote the parallel translation of aⁱ4fⁱia on the factor manifolds Mⁱ.

R E F E R E N C E S

[1] G. GALANIS,On a type of linear differential equations in Fréchet spaces, An- nali della Scuola Normale Superiore di Pisa, 4, No. 24 (1997), pp. 501- 510.

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Manoscritto pervenuto in redazione l’8 novembre 2003.